Biryukov equation

{{short description|Non-linear second-order differential equation}}

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{{context|date=May 2025}}

{{technical|date=May 2025}}

{{COI|date=January 2017}}

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File:Sine oscillations.jpg oscillations {{math|1=F = 0.01}}]]

In the study of dynamical systems, the Biryukov equation (or Biryukov oscillator), named after Vadim Biryukov (1946), is a non-linear second-order differential equation used to model damped oscillators.H. P. Gavin, The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems (MATLAB implementation included)

The equation is given by

$$\backslash frac\{d^2\; y\}\{dt^2\}+f(y)\backslash frac\{dy\}\{dt\}+y=0,\; \backslash qquad\backslash qquad\; (1)$$

where {{math|ƒ(y)}} is a piecewise constant function which is positive, except for small {{mvar|y}} as

$$\backslash begin\{align\}$$

& f(y) = \begin{cases} -F, & |y|\le Y_0; \\[4pt] F, & |y|>Y_0. \end{cases} \\[6pt]

& F = \text{const.} > 0, \quad Y_0 = \text{const.} > 0.

\end{align}

Eq. (1) is a special case of the Lienard equation; it describes the auto-oscillations.

Solution (1) at separate time intervals when f(y) is constant is given byArrowsmith D. K., Place C. M. Dynamical Systems. Differential equations, maps and chaotic behavior. Chapman & Hall, (1992)

$$y\_k(t)\; =\; A\_\{1,k\}\backslash exp(s\_\{1,k\}t)\; +\; A\_\{2,k\}\backslash exp(s\_\{2,k\}t)\; \backslash qquad\backslash qquad\; (2)$$

where {{math|exp}} denotes the exponential function. Here

$$s\_k\; =\; \backslash begin\{cases\}$$

\displaystyle \frac{F}{2}\mp\sqrt{ \left(\frac{F}{2}\right)^2-1}, & |y|

\displaystyle -\frac{F}{2}\mp\sqrt{ \left(\frac{F}{2}\right)^2-1} & \text{otherwise.}

\end{cases}

Expression (2) can be used for real and complex values of {{mvar|s{{sub|k}}}}.

The first half-period’s solution at $y(0)=\backslash pm\; Y\_0$ is File:Fig 2 Relaxation oscillations F = 4.png

$$\backslash begin\{align\}$$

y(t) &= \begin{cases}

y_1(t), & 0\le t

y_2(t), & \displaystyle T_0\le t< \frac{T}{2}. \end{cases} \\[4pt]

y_1(t) &= A_{1,k}\cdot \exp (s_{1,k}t)+A_{2,k}\cdot \exp (s_{2,k}t), \\[2pt]

y_2(t) &= A_{3,k}\cdot \exp(s_{3,k}t)+A_{4,k}\cdot \exp (s_{4,k}t).

\end{align}

The second half-period’s solution is

$$y(t)=\; \backslash begin\{cases\}$$

\displaystyle -y_1\left(t-\frac{T}{2}\right), & \displaystyle \frac{T}{2} \le t < \frac{T}{2} + T_0; \\[4pt]

\displaystyle -y_2\left(t-\frac{T}{2}\right), & \displaystyle \frac{T}{2} + T_0 \le t < T. \end{cases}

The solution contains four constants of integration {{math|A{{sub|1}}, A{{sub|2}}, A{{sub|3}}, A{{sub|4}}}}, the period {{mvar|T}} and the boundary {{math|T{{sub|0}}}} between {{math|y{{sub|1}}(t)}} and {{math|y{{sub|2}}(t)}} needs to be found. A boundary condition is derived from the continuity of {{math|y(t)}} and {{math|dy/dt}}.Pilipenko A. M., and Biryukov V. N. «Investigation of Modern Numerical Analysis Methods of Self-Oscillatory Circuits Efficiency», Journal of Radio Electronics, No 9, (2013). http://jre.cplire.ru/jre/aug13/9/text-engl.html

Solution of (1) in the stationary mode thus is obtained by solving a system of algebraic equations as

$$\backslash begin\{array\}\{ll\}$$

& y_1(0) = -Y_0 & y_1(T_0) = Y_0 \\[6pt]

& y_2(T_0) = Y_0 & y_2 \! \left(\tfrac{T}{2}\right) = Y_0 \\[6pt]

& \displaystyle \left.\frac{dy_1}{dt}\right|_{T_0} = \left.\frac{dy_2}{dt}\right|_{T_0} \qquad

& \displaystyle \left.\frac{dy_1}{dt}\right|_{0} = -\left.\frac{dy_2}{dt}\right|_\frac{T}{2}

\end{array}

The integration constants are obtained by the Levenberg–Marquardt algorithm.

With $f(y)=\backslash mu(-1+y^2)$, $\backslash mu\; =\; \backslash text\{const.\}\; >\; 0,$ Eq. (1) named Van der Pol oscillator. Its solution cannot be expressed by elementary functions in closed form.